Every LWF and AMP chain graph originates from a set of causal models

This paper aims at justifying LWF and AMP chain graphs by showing that they do not represent arbitrary independence models. Specifically, we show that every chain graph is inclusion optimal wrt the intersection of the independence models represented by a set of directed and acyclic graphs under conditioning. This implies that the independence model represented by the chain graph can be accounted for by a set of causal models that are subject to selection bias, which in turn can be accounted for by a system that switches between different regimes or configurations.